channel - Advances in Electronics and Telecommunications

BACCI AND LUISE: GAME THEORY IN WIRELESS COMMUNICATIONS WITH AN APPLICATION TO SIGNAL SYNCHRONIZATION 87
player 1
AP
Fig. 1. The network scenario in the near-far effect game.
player 2
players - the so-called the Nash equilibrium [27], [28]), game
theory has been a subject of considerable study, especially in
the traditional areas of economics and political science.
More recently, game theory has been also used in telecommunications
and wireless communications [29]–[36]. The reason
for this blooming of game-theoretic applications lies in
the nature of typical interactions between users in a wireless
network. The wireless terminals can in fact be modeled as
players in a game competing for the network resources (e.g.,
bandwidth and power), which are typically scarce. Any action
taken by a user affects the performance of other users as
well. Thus, game theory turns out to be a natural tool for
investigating this interplay.
In the next subsections, we will provide some motivating
examples for its use in the context of power control for
multiaccess wireless networks, also introducing some key
definitions and the fundamental analytical tools.
A. Noncooperative games
A (strategic) game consists of three components: a set of
players, the strategy set for each player, and a utility (payoff)
for each player measuring its level of satisfaction [37]. In
its mathematical formulation, the game can be represented
as G = [K, {Ak}, {uk (a)}], where K = {1, . . . , K} is the
set of players; Ak is the set of actions (strategies) available
to player k; and uk (a) is the utility (payoff) for player k.
The utility depends not only on its own strategy ak ∈ Ak,
but also on the actual strategies taken by all of the other
players, denoted by a \k = (a1, . . . , ak−1, ak+1, . . . , aK).
Hence, uk (a) = uk
� �
ak, a \k .
Since our focus is on distributed schemes, we concentrate
on noncooperative games, where each player k chooses its
strategy a ∗ k to unilaterally maximize its own utility uk (a):
a ∗ k
= arg max uk (a)
ak∈Ak
= arg max uk
ak∈Ak
� �
ak, a \k , (1)
where the latter notation emphasizes that the k-th player has
control over its own strategy ak only. In other terms, a ∗ k
represents player k’s best response to the concurrent actions of
the other players. For these games, we first need to introduce
two fundamental concepts, namely, Nash equilibrium and
Pareto optimality [37].
Definition 1: A Nash equilibrium (NE) is a set of strategies
a∗ � �
= such that no user can unilaterally improve its
a ∗ k , a∗ \k
near player
(player 1)
0
p
0
far player
(player 2)
0, 0 0, 1 − c
1 − c, 0 1 − c, −c
p
u1 (p1, p2) , u2 (p1, p2)
Fig. 2. Payoff matrix for the near-far effect game.
own utility, i.e.,
�
uk a ∗ k , a∗ �
\k ≥ uk
�
ak, a ∗ �
\k
∀ak ∈ Ak, k ∈ K, (2)
where obviously a∗ \k = � a∗ 1, . . . , a∗ k−1 , a∗ k+1 , . . . , a∗ �
K . In
other words, an NE is a stable outcome of a game in which
multiple agents with conflicting interests compete through
self-optimization and reach a point where no player has any
incentive to unilaterally deviate (whence stability).
Definition 2: A set of strategies ã = � �
ãk, ã \k is Paretooptimal
if there exists no other set a = � �
� � � � ak, a \k�
such � that
uk ak, a \k ≥ uk ãk, ã \k for all k ∈ K and uk
� � ak, a \k >
uk ãk, ã \k for some k ∈ K. In other words, when all players
settle onto a Pareto-optimal strategy, no player can improve
its own utility without reducing the utility of at least another
player.
Our focus throughout this work is on pure (i.e., deterministic)
strategies. However, players can also opt for mixed (i.e.,
statistical) strategies. In this case, each player chooses its
strategy according to a probability distribution that is known
to the other players. Nash proved that a finite noncooperative
game always has at least one mixed-strategy NE [27], [28].
This means that a noncooperative game may have no purestrategy
equilibria, one pure-strategy equilibrium, or multiple
pure-strategy equilibria. We can also easily show that there
could exist more than one Pareto-optimal solution, and that,
in general, an NE does not correspond to a Pareto-optimal
strategy.
To illustrate the intuitive meaning of these concepts, we
consider a trivial example of a static1 noncooperative game,
that we call the near-far effect game. Two wireless terminals
(player 1 and player 2) transmit to a certain access point (AP)
in a CDMA network. Player 1 is located close to the AP,
whilst player 2 is much farther away, as is depicted in Fig. 1.
Hence, K = 2 and K = {1, 2}. Each user is allowed either to
transmit at a certain power level pk = p, or to wait (pk = 0).
This translates into Ak = pk = {0, p}. Each terminal achieves
a degree of satisfaction which depends on the outcome of the
transmission and on the expenditure in terms of cost of the
1 A game is said to be static if there exists only one time step, which means
that the players’ strategies are carried out through a single move [37].